On 2/21/2013 2:08 AM, Shmuel (Seymour J.) Metz wrote:>> You have to start with a topology to have a quotient topology. There> is no need to model theories on topological spaces.>

All Boolean-valued forcing models are based on topologiesconstructed from regular open sets in a complete Booleanalgebra. These topologies are semi-regular.

My point is that the very syntax of a first-order languagecan be recognized as a minimal Hausdorff topology as soonas one places a mutually exclusive bivalent truth functionalityonto its symbols. Minimal Hausdorff topologies aresemi-regular.

Moreover, you cannot divorce this structure from a logicintended as a deductive calculus because what makes itinterpretable as a deductive calculus is its relationshipto the truth-conditions of interpretations.